Fixed point results for generalized multi-valued contractions

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1 Available online at J. Nonlinear Sci. Appl. 8 (2015), Research Article Fixed point results for generalized multi-valued contractions Jamshaid Ahmad a,, Nawab Hussain b, Abdul Rahim Khan c, Akbar Azam a a Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. b Department of Mathematics, King Abdulaziz University, P.O. Box 8020, Jeddah 21589, Saudi Arabia. c Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 1261, Saudi Arabia Abstract Javahernia et al. [Fixed Point Theory and Applications 2014, 2014:195] introduced the concept of generalized Mizoguchi-Takahashi type contractions and established some common fixed point results for such contractions. In this paper, we define the notion of generalized α Mizoguchi-Takahashi type contractions and obtain some new fixed point results which generalize various results existing in literature. An example is included to show that our results are genuine generalization of the corresponding known results. c 2015 All rights reserved. Keywords: Metric space, fixed point, MT-function 2010 MSC: 47H10, 54H Introduction and Preliminaries Let (X, d) be a metric space. For x X and A X, we denote d(x, A) = inf{d(x, y) : y A}. Let us denote by N(X), the class of all nonempty subsets of X, CL(X), the class of all nonempty closed subsets of X, CB(X), the class of all nonempty closed and bounded subsets of X and K(X), the class of all compact subsets of X. Let H be the Hausdorff-Pompeiu metric induced by metric d on X, that is, H(A, B) = max{supd(x, B), supd(y, A)} x A y B for every A, B CB(X). Let S : X CL(X) be a multivalued mapping. A point q X is said to be a fixed point of S if q Sq. If, for x 0 X, there exists a sequence {x n } n N in X such that x n Sx n 1, then Corresponding author addresses: jamshaid_jasim@yahoo.com (Jamshaid Ahmad), nhusain@kau.edu.sa (Nawab Hussain), arahim@kfupm.edu.sa (Abdul Rahim Khan), akbarazam@yahoo.com (Akbar Azam) Received

2 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), the orbit of S is defined as O(S, x 0 ) = {x 0, x 1, x 2,...}. A mapping g : X R is said to be S-orbitally lower semi-continuous if {x n } n N is a sequence in O(S, x 0 ) and x n υ implies g(υ) lim n inf g(x n ). Nadler [16] extended the Banach contraction principle to multivalued mappings in the following way. Theorem 1.1 ([16]). Let (X, d) be a complete metric space and S : X CB(X) be a multivalued mapping such that for all x, y X, where 0 k < 1. Then S has a fixed point. H(S(x), S(y)) kd(x, y) (1.1) Reich [18], established the following fixed point theorem for the case of multivalued mappings with compact range. Theorem 1.2 ([18]). Let (X, d) be a complete metric space and ϕ : [0, ) [0, 1) be such that lim sup ϕ(v) < 1 v u + for each u [0, ). If S : X K(X) is a multivalued mapping satisfying for all x, y X, then S has a fixed point. H(S(x), S(y)) ϕ(d(x, y))d(x, y) (1.2) An open problem posed by Reich [18] asks wether the above theorem holds for mapping S : X CB(X). Mizoguchi and Takahashi [15] proved the following famous result as a generalization of Nadler s fixed point theorem [16]. Theorem 1. ([15]).. mapping. Assume that Let (X, d) be a complete metric space and S : X CB(X) be a multivalued for all x, y X, where ϕ : [0, ) [0, 1) is such that for each u [0, ). Then S has a fixed point. H(S(x), S(y)) ϕ(d(x, y))d(x, y) (1.) lim sup ϕ(v) < 1 v u + The above function ϕ : [0, ) [0, 1) of Mizoguchi Takahashi is named as MT-function. As in [15], we denote by Ω the set of all functions ϕ : [0, ) [0, 1). Kamran [12], generalized Mizoguchi and Takahashi s theorem in the following way. Theorem 1.4 ([12]). Let (X, d) be a complete metric space, ϕ : [0, ) [0, 1) be an MT- function and S : X CL(X) a multivalued mapping. Assume that for each x X and y Sx where ϕ Ω. Then d(y, S(y)) ϕ(d(x, y))d(x, y) (1.4) Asl et al. [2] defined the notion of α -admissible mappings as follows:

3 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Definition 1.5. ([2]). Let (X, d) be a metric space, α : X X [0, + ) and S : X CL(X) be given. We say that S is α admissible if for all x X and y Sx with α(x, y) 1, we have that α (Sx, Sy) 1, where α (Sx, Sy) = inf{α(a, b) : a Sx, b Sy}. Kiran et al. [1] generalized Theorem 1.4 of Kamran [12] for α -admissible mapping as follows. Theorem 1.6 ([1]). Let (X, d) be a complete metric space and S : X CL(X) be α admissible such that α (S(x), S(y))d(y, S(y)) ϕ(d(x, y))d(x, y) (1.5) x X with y Sx and ϕ Ω. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then Very recently, Javahernia et al. [10] introduced the concept of generalized Mizoguchi Takahashi function and proved some new common fixed point results. Definition 1.7. ([10]). A function β : R R R is said to be generalized Mizoguchi Takahashi function if the following conditions hold: (a 1 ) 0 < β(u, v) < 1 for all u, v > 0; (a 2 ) for any bounded sequence {u n } (0, + ) and any non-increasing sequence {v n } (0, + ), we have lim sup β(u n, v n ) < 1. n Consistent with Javahernia et al. [10], we denote by Λ the set of all functions β : R R R satisfying the conditions (a 1 ) (a 2 ). They gave the following example of a generalized Mizoguchi Takahashi function. Example 1.8. Let m(u) = ln(u+10) u+9 for all u > 9. Define Then β Λ. β(t, s) = { t s 2 +1, 1 < t < s, m(s), otherwise. For more details, we refer the reader to [1,, 4, 5, 6, 7, 8, 9, 11, 14, 17, 19, 20]. In this paper, motivated by Javahernia et al. [10], we establish some new fixed point theorems. Our new results generalize and improve fixed point theorems due to Kiran-Ali-Kamran [1], Mizoguchi-Takahashi [15] and Nadler [16]. The following lemma is crucial for the proofs of our results. Lemma 1.9 ([12]). Let (X, d) be a metric space and B nonempty, closed subset of X and q > 1. Then, for each x X with d(x, B) > 0 and q > 1, there exists b B such that d(x, b) < qd(x, B). Throughout this article, N, R +, R stand for the set of: natural numbers, positive real numbers and real numbers, respectively.

4 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Main Results We start this section with the definition of generalized α Mizoguchi-Takahashi type contraction. Definition 2.1. Let (X, d) be a metric space. The mapping S : X CL(X) is said to be generalized α Mizoguchi-Takahashi type contraction if there exist functions α : X X [0, + ) and β Λ such that α (S(x), S(y))d(y, S(y)) β(d(y, S(y)), d(x, y))d(x, y) (2.1) for all x X, with y Sx. Here is our main result. Theorem 2.2. Let (X, d) be a complete metric space and S : X CL(X) be generalized α Mizoguchi- Takahashi type contraction and α admissible. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then Proof. By hypothesis, we have x 0 X and x 1 Sx 0 with α(x 0, x 1 ) 1. As S is α admissible, so we have α (Sx 0, Sx 1 ) 1. If x 0 = x 1, then x 0 is fixed point of S. Let x 0 x 1 i.e d(x 0, x 1 ) > 0. If x 1 Sx 1, then x 1 is fixed point of S. Assume that x 1 Sx 1, that is, d(x 1, S(x 1 )) > 0. Since d(x 0, x 1 ) > 0 and d(x 1, S(x 1 )) > 0 1 so by taking h = > 1, it follows by Lemma 1.9, that there exists x 2 Sx 1 such that β(d(x1,s(x 1 )),d(x 0,x 1 )) d(x 1, x 2 ) d(x 1, S(x 1 )) β(d(x1, S(x 1 )), d(x 0, x 1 )) α (2.2) (S(x 0 ), S(x 1 ))d(x 1, S(x 1 )) β(d(x1, S(x 1 )), d(x 0, x 1 )). From (2.1), we have α (S(x 0 ), S(x 1 ))d(x 1, S(x 1 )) β(d(x 1, S(x 1 )), d(x 0, x 1 ))d(x 0, x 1 )). (2.) As S is α admissible, so α(x 1, x 2 ) α (S(x 0 ), S(x 1 )) 1 implies α (S(x 1 ), S(x 2 )) 1. If x 1 = x 2, then x 1 is fixed point of S. Let x 1 x 2 i.e d(x 1, x 2 ) > 0. If x 2 S(x 2 ), then x 2 is fixed point of S. Assume that x 2 Sx 2, that is, d(x 2, S(x 2 )) > 0. Since d(x 1, x 2 ) > 0 and d(x 2, S(x 2 )) > 0 so by taking 1 h = > 1, it follows by Lemma 1.9, that there exists x Sx 2 such that β(d(x2,s(x 2 )),d(x 1,x 2 )) d(x 2, x ) d(x 2, S(x 2 )) β(d(x2, S(x 2 )), d(x 1, x 2 )) α (2.4) (S(x 1 ), S(x 2 ))d(x 2, S(x 2 )) β(d(x2, S(x 2 )), d(x 1, x 2 )). From (2.1), we have α (S(x 1 ), S(x 2 ))d(x 2, S(x 2 )) β(d(x 2, S(x 2 )), d(x 1, x 2 ))d(x 1, x 2 )). (2.5) Repeating the above procedure, we obtain a sequence {x n } n N in X such that x n Sx n 1, α (Sx n 1, Sx n ) 1 for each n N and

5 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), d(x n, x n+1 ) d(x n, S(x n )) β(d(xn, S(x n )), d(x n 1, x n )) α (S(x n 1 ), S(x n ))d(x n, S(x n )) β(d(xn, S(x n )), d(x n 1, x n )) (2.6) for all n = 1, 2,... We have assumed that x n 1 x n, otherwise x n 1 is a fixed point of S. Also x n Sx n for all n = 1, 2,... From (2.1), we have α (S(x n 1 ), S(x n ))d(x n, S(x n )) β(d(x n, S(x n )), d(x n 1, x n ))d(x n 1, x n )) (2.7) for all n = 1, 2,..., which implies that {d(x n, Sx n )} n N is a bounded sequence. Combining (2.6) and (2.7), we have d(x n, x n+1 ) β(d(x n, S(x n )), d(x n 1, x n ))d(x n 1, x n ) < d(x n 1, x n ) (2.8) for n = 1, 2,... It means that {d(x n 1, x n )} n N is a strictly decreasing sequence of positive real numbers. So lim d(x n, x n+1 ) = inf d(x n, x n+1 ) = l. (2.9) n n N By (a 2), we have lim sup β(d(x n, Sx n ), d(x n 1, x n )) < 1. (2.10) n Now we claim that l = 0. Otherwise, by taking limit in (2.8), we get which is a contradiction. Hence l lim n sup β(d(x n, Sx n ), d(x n 1, x n ))l < l lim d(x n, x n+1 ) = inf d(x n, x n+1 ) = 0. (2.11) n n N Now we prove that {x n } n N is a Cauchy sequence in X. For each n N, let λ n := β(d(x n, Sx n ), d(x n 1, x n )). Then λ n (0, 1) for all n N. By (2.8), we obtain d(x n, Sx n ) α (S(x n 1 ), S(x n ))d(x n, S(x n )) β n d(x n 1, x n ) (2.12) for all n N. By (a 2 ), we have lim n sup λ n < 1, so there exist c [0, 1) and n 0 N, such that Thus for any n n 0, from (2.12) and (2.1), we have λ n c, for all n N with n n 0. (2.1) d(x n, x n+1 ) λ n d(x n 1, x n ) λ n λ n 1 d(x n 2, x n 1 ). λ n λ n 1 λ n 2...λ n0 d(x 0, x 1 ) c n n 0+1 d(x 0, x 1 ). Put δ n = cn n c d(x 0, x 1 ) for n N and n n 0. For m, n N with m > n n 0, we have d(x n, x m ) d(x n, x n+1 ) + d(x n+1, x n+2 ) d(x m 1, x m ) c n n 0+1 d(x 0, x 1 ) + c n n 0+2 d(x 0, x 1 ) c n n 0+m d(x 0, x 1 ) c n n 0+1 (1 + c + c c m 1 )d(x 0, x 1 ) δ n. (2.14)

6 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Since c [0, 1), lim n δ n = 0 and hence lim n sup{d(x n, x m ) : m > n} = 0. Thus {x n } n N is a Cauchy sequence in X. Since X is complete so there exists w X such that x n w. Since x n Sx n 1, it follows from (2.12) that d(x n, Sx n ) α (S(x n 1 ), S(x n ))d(x n, Sx n ) β(d(x n, Sx n ), d(x n 1, x n ))d(x n 1, x n ). (2.15) Letting n + in (2.15), we have Suppose g(x) = d(x n, Sx n ) is S-orbitally lower continuous at u. Then lim d(x n, Sx n ) = 0. (2.16) n d(w, Sw) = g(w) lim n inf g(x n) = lim n inf d(x n, Sx n ) = 0. Since Sw is closed, so w Sw. Conversely, if w is fixed point of S, then g(w) = 0 lim n inf g(x n). The proofs of the following theorems are similar to the proof of Theorem 2.2 and so are omitted. Theorem 2.. Let (X, d) be a complete metric space and S : X CL(X) be α admissible such that α (y, S(y))d(y, S(y)) β(d(y, S(y)), d(x, y))d(x, y) (2.17) for all x X, with y Sx and β Λ. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then Theorem 2.4. Let (X, d) be a complete metric space and S : X CL(X) be α admissible such that α(x, y)d(y, S(y)) β(d(y, S(y)), d(x, y))d(x, y) (2.18) for all x X, with y Sx and β Λ. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then. Example In this section, we construct an example which shows that Theorem 2.2 is a proper generalization of Theorem 1.6. Example.1. Let X = [0, 1] and d : X X R be the usual metric. Define S : X CL(X) by and α : X X [0, + ) by { 1 S(x) = x2, for x [0, 6 11 ) ( 6 11, 1] { } for x = 6 11 { 1, if x, y [0, 6 α(x, y) = 11 ) ( 6 11, 1] 0 otherwise.

7 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Define ϕ : [0, ) [0, 1) by ϕ(t) = 4 9 x, for x [0, 10 ) ( 10, 1 ) 9 11 for x = 10 1 for x [ 1, ) and β : R R R by β(u, v) = 1 ϕ(v) v for all u, v > 0. For any bounded sequence {u n } (0, + ) and any non-increasing sequence {v n } (0, + ), we have lim sup β(u n, v n ) = lim sup(1 ϕ(v n) ) < 1. n n v n We show that S satisfies all the hypotheses of our Theorem 2.2. It is easy to see that the function g(x) = d(x, S(x)) is lower semi-continuous. Moreover, for each x [0, 6 11 ) ( 6 11, 1], we have S(x) = { 1 x2 } and therefore y = 1 x2. Now d(x, y) = d(x, S(x)) = x 1 x2. Further, α (S(x), S(y))d(y, S(y)) = d( 1 x2, 1 27 x4 ) = 1 (x2 ( 1 x2 ) 2 ) = 1 (x + 1 x2 )(x 1 x2 ) 5 d(x, y) 9 = β(d(y, S(y)), d(x, y))d(x, y). Take x = Then we have have S(x) = { 110 } and d(x, y) = d(x, S(x)) = 10. The contractive condition (2.1) is satisfied trivially. Now we show that a given map S does not satisfy hypotheses of Theorem 1.6 of Kiran et al. [1]. For x = 1, we have we have S(x) = { 1 }, y = 1 1 and S(y) = { 27 }. Then d(x, y) = 2 8 and d(y, S(y)) = 27. Consequently, α (S(x), S(y))d(y, S(y)) = > 1.2 = ϕ(d(x, y))d(x, y). Therefore, for x = 1, the inequality (1.5) in Theorem 1.6 is not satisfied. 4. Consequences Remark 4.1. Theorem 2.2 improves Theorem 1.6, since S may take values in CL(X) and d(y, S(y)) H(S(x), S(y)) for y S(x). Corollary 4.2. Let (X, d) be a complete metric space and S : X CL(X) be α admissible such that α (S(x), S(y))H(S(x), S(y)) β(d(y, S(y)), d(x, y))d(x, y) for each x X and y Sx where β Λ. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then Corollary 4.. Theorem 1.6 follows from Theorem 2.2 by putting β(u, v) = ϕ(v). Remark 4.4. Taking β(u, v) = ϕ(v) in Theorem 2. and Theorem 2.4, we obtain Theorem 2.6 and Theorem 2.7 in [1], respectively.

8 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Corollary 4.5. Let (X, d) be a complete metric space and S : X CL(X) satisfies for each x X and y Sx where β Λ. Then d(y, S(y)) β(d(y, S(y)), d(x, y))d(x, y) Proof. Define α : X X [0, + ) by α(x, y) = 1 for each x, y X. Then the proof follows from Theorem 2.2. Remark 4.6. Taking β(u, v) = ϕ(v) in Corollary 4.5, we can get Theorem 1.6 which is Theorem 2.1 in [12]. Corollary 4.7. Let (X, d) be a complete metric space and S : X CL(X) satisfies for all x X, with y Sx where µ [0, 1). Then d(y, S(y)) µd(x, y) Proof. Take β(u, v) = µ and apply Corollary 4.5. Corollary 4.8. Let (X, d) be a complete metric space and S : X CL(X) satisfies d(y, S(y)) ϕ(d(x, y)) for each x X and y Sx where ϕ : [0, ) [0, 1) is a function such that ϕ(v) < v and lim v u + sup ϕ(v) v < 1. Then Proof. Take β(u, v) = ϕ(v) v and apply Corollary 4.5. Javahernia et al. [10] also introduced the concept of weak l.s.c. in the following way. Definition 4.9. A function φ : [0, ) [0, ) is said to be weak l.s.c. sequence {u n } (0, + ), we have function if for each bounded lim inf φ(u n) > 0. n Consistent with Javahernia et al. [10], we denote by Ϝ, the set of all functions φ : [0, ) [0, ) satisfying the above condition.

9 J. Ahmad, N. Hussain, A. R. Khan, A. Azam, J. Nonlinear Sci. Appl. 8 (2015), Theorem Let (X, d) be a complete metric space and S : X CL(X) be α admissible such that α (S(x), S(y))d(y, S(y)) d(x, y) φ(d(x, y)) for each x X and y Sx where φ : [0, ) [0, ) is such that φ(0) = 0, φ(v) < v and φ Ϝ. Suppose that there exist x 0 X and x 1 Sx 0 such that α(x 0, x 1 ) 1. Then Proof. Define β(u, v) = 1 φ(u) u for all u, v > 0. Since for each bounded sequence {u n } (0, + ), we have lim n inf φ(u n ) > 0. So lim n inf φ(un) u n > 0. Thus This shows that β Λ. Also Thus by Theorem 2.2, w is fixed point of S. lim sup(1 φ(u n) ) = 1 lim inf φ(u n) < 0. n n u n u n α (S(x), S(y))d(y, S(y)) β(d(y, S(y)), d(x, y))d(x, y). Corollary Let (X, d) be a complete metric space and S : X CL(X) be such that d(y, S(y)) d(x, y) φ(d(x, y)) for each x X and y Sx, where φ : [0, ) [0, ) is such that φ(0) = 0, φ(v) < v and φ Ϝ. Then Proof. Define α : X X [0, + ) by α(x, y) = 1 for each x, y X. Then the proof follows from Theorem References [1] J. Ahmad, A. Al-Rawashdeh, A. Azam, Fixed point results for {α, ξ}-expansive locally contractive mappings, J. Inequal. Appl., 2014 (2014), 10 pages. 1 [2] J. H. Asl, S. Rezapour, N. Shahzad, On fixed points of α ψ contractive multifunctions, Fixed Point Theory Appl., 2012 (2012), 6 pages.1, 1.5 [] A. Azam, M. Arshad, Fixed points of a sequence of locally contractive multivalued maps, Comp. Math. Appl., 57 (2009), [4] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math. (1922), [5] L. Ćirić, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput., 217 (2011), [6] N. Hussain, P. Salimi, A. Latif, Fixed point results for single and set-valued α -η -ψ-contractive mappings, Fixed Point Theory Appl., 201 (201), 2 pages. 1 [7] N. Hussain, J. Ahmad, A. Azam, Generalized fixed point theorems for multivalued α ψ contractive mappings, J. Ineq. Appl., 2014 (2014), 15 pages.1 [8] N. Hussain, N. Yasmin, N. Shafqat, Multi-valued Ćirić contractions on metric spaces with applications, Filomat., 28 (2014), [9] N. Hussain, V. Parvaneh, S. J. Hoseini Ghoncheh, Generalized contractive mappings and weakly α-admissible pairs in G-metric spaces, The Scientific World J., 2014 (2014), 15 pages. 1

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